5/1 - Revision history

Overthink: /* See also */ link 5/4

2026-03-20T15:38:06Z

Overthink: + prime harmonic

2026-03-19T05:40:25Z

+ prime harmonic

← Older revision		Revision as of 05:40, 19 March 2026
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	}}		}}
	'''5/1''', the '''5th harmonic''', '''pentave''' or '''quintuple''', is the [[harmonic]] past [[4/1]] and before [[6/1]]. It is two [[octave]]s above [[5/4]], and is the basis of [[5-limit]] harmony, as many 5-limit intervals can be expressed as the difference between this and another harmonic.		'''5/1''', the '''5th harmonic''', '''pentave''' or '''quintuple''', is the [[harmonic]] past [[4/1]] and before [[6/1]]. It is two [[octave]]s above [[5/4]], and is the basis of [[5-limit]] harmony, as many 5-limit intervals can be expressed as the difference between this and another harmonic.

			5/1 is the third [[prime harmonic]], after [[3/1]] and before [[7/1]].

	5/1 is on a list of integer harmonics that approximate closest a given stack of fifths, the error being the [[81/80\|syntonic comma]].<ref>{{OEIS\|A081464}} – Numbers ''k'' such that the fractional part of (3/2)<sup>''k''</sup> decreases monotonically to zero</ref><ref>{{OEIS\|A267122}} – Numbers ''n'' such that (3/2)<sup>n</sup> is closer to an integer than (3/2)<sup>m</sup> for any 0 < ''m'' < ''n''</ref>		5/1 is on a list of integer harmonics that approximate closest a given stack of fifths, the error being the [[81/80\|syntonic comma]].<ref>{{OEIS\|A081464}} – Numbers ''k'' such that the fractional part of (3/2)<sup>''k''</sup> decreases monotonically to zero</ref><ref>{{OEIS\|A267122}} – Numbers ''n'' such that (3/2)<sup>n</sup> is closer to an integer than (3/2)<sup>m</sup> for any 0 < ''m'' < ''n''</ref>

Overthink: ratios, not steps

2026-03-16T23:00:30Z

ratios, not steps

← Older revision		Revision as of 23:00, 16 March 2026
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	5/1 is the first prime harmonic greater than [[3/1\|3]], and is thus the first that adds [[interval quality\|interval qualities]] other than those found in [[Pythagorean tuning]]. Simple ratios of 5 tend to differ from Pythagorean intervals by the syntonic comma [[81/80]] (21.5{{C}}), which contains a single factor of 5 in its denominator. In [[diatonic]] interval classification, simple ratios with a factor of 5 in the numerator, such as [[5/4]], [[10/9]], and [[15/8]], are major intervals, while simple ratios with a factor of 5 in the denominator, such as [[6/5]], [[16/15]], and [[9/5]], are minor intervals. Major intervals are flat of their corresponding Pythagorean interval by 81/80; for example, 5/4 is flat of [[81/64]], the Pythagorean major third, by 81/80. Minor intervals, on the other hand are sharp of their corresponding Pythagorean interval by 81/80; for example, 6/5 is 81/80 sharp of [[32/27]], the Pythagorean minor third. A simple ratio of 5 iscan generally more [[consonant]] than the Pythagorean interval which was modified by 81/80 to reach it; for example, 5/4 is more consonant than 81/64, and 6/5 is more consonant than 32/27. [[Meantone]] tempers out 81/80 so that the Pythagorean and 5-limit thirds are equated.		5/1 is the first prime harmonic greater than [[3/1\|3]], and is thus the first that adds [[interval quality\|interval qualities]] other than those found in [[Pythagorean tuning]]. Simple ratios of 5 tend to differ from Pythagorean intervals by the syntonic comma [[81/80]] (21.5{{C}}), which contains a single factor of 5 in its denominator. In [[diatonic]] interval classification, simple ratios with a factor of 5 in the numerator, such as [[5/4]], [[10/9]], and [[15/8]], are major intervals, while simple ratios with a factor of 5 in the denominator, such as [[6/5]], [[16/15]], and [[9/5]], are minor intervals. Major intervals are flat of their corresponding Pythagorean interval by 81/80; for example, 5/4 is flat of [[81/64]], the Pythagorean major third, by 81/80. Minor intervals, on the other hand are sharp of their corresponding Pythagorean interval by 81/80; for example, 6/5 is 81/80 sharp of [[32/27]], the Pythagorean minor third. A simple ratio of 5 iscan generally more [[consonant]] than the Pythagorean interval which was modified by 81/80 to reach it; for example, 5/4 is more consonant than 81/64, and 6/5 is more consonant than 32/27. [[Meantone]] tempers out 81/80 so that the Pythagorean and 5-limit thirds are equated.

	The [[octave-reduced]] 5th harmonic, the classical major third 5/4, combines nicely with the octave-reduced 1st and 3rd harmonics, being the unison and perfect fifth with ratios [[1/1]] and [[3/2]] respectively, to form the major triad [[4:5:6\|1–5/4–3/2]] (4:5:6), which consists of the 5/4 major third and the 6/5 minor third stacked on top of each other, and is [[otonal]]. Its inverse is the minor triad [[10:12:15\|1–6/5–3/2]] (1/(6:5:4) = 10:12:15), which consists of 6/5 and 5/4 stacked on top of each other, and is [[utonal]]. More extended chords can be built with these intervals; for example the [[dominant seventh chord]] with ~~steps~~ [[20:25:30:36\|1–5/4–3/2–9/5]] (20:25:30:36), which is widely used in classical and contemporary music.		The [[octave-reduced]] 5th harmonic, the classical major third 5/4, combines nicely with the octave-reduced 1st and 3rd harmonics, being the unison and perfect fifth with ratios [[1/1]] and [[3/2]] respectively, to form the major triad [[4:5:6\|1–5/4–3/2]] (4:5:6), which consists of the 5/4 major third and the 6/5 minor third stacked on top of each other, and is [[otonal]]. Its inverse is the minor triad [[10:12:15\|1–6/5–3/2]] (1/(6:5:4) = 10:12:15), which consists of 6/5 and 5/4 stacked on top of each other, and is [[utonal]]. More extended chords can be built with these intervals; for example the [[dominant seventh chord]] with ratios [[20:25:30:36\|1–5/4–3/2–9/5]] (20:25:30:36), which is widely used in classical and contemporary music.

	== See also ==		== See also ==

Overthink: Add a section

2026-03-16T23:00:09Z

Add a section

← Older revision		Revision as of 23:00, 16 March 2026
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	'''5/1''', the '''5th harmonic''', '''pentave''' or '''quintuple''', is the [[harmonic]] past [[4/1]] and before [[6/1]]. It is two [[octave]]s above [[5/4]], and is the basis of [[5-limit]] harmony, as many 5-limit intervals can be expressed as the difference between this and another harmonic.		'''5/1''', the '''5th harmonic''', '''pentave''' or '''quintuple''', is the [[harmonic]] past [[4/1]] and before [[6/1]]. It is two [[octave]]s above [[5/4]], and is the basis of [[5-limit]] harmony, as many 5-limit intervals can be expressed as the difference between this and another harmonic.

	5/1 is on a list of integer harmonics that approximate closest a given stack of fifths, the error being the [[81/80\|syntonic comma]].		5/1 is on a list of integer harmonics that approximate closest a given stack of fifths, the error being the [[81/80\|syntonic comma]].<ref>{{OEIS\|A081464}} – Numbers ''k'' such that the fractional part of (3/2)<sup>''k''</sup> decreases monotonically to zero</ref><ref>{{OEIS\|A267122}} – Numbers ''n'' such that (3/2)<sup>n</sup> is closer to an integer than (3/2)<sup>m</sup> for any 0 < ''m'' < ''n''</ref>

			== Intervals of 5 ==
			5/1 is the first prime harmonic greater than [[3/1\|3]], and is thus the first that adds [[interval quality\|interval qualities]] other than those found in [[Pythagorean tuning]]. Simple ratios of 5 tend to differ from Pythagorean intervals by the syntonic comma [[81/80]] (21.5{{C}}), which contains a single factor of 5 in its denominator. In [[diatonic]] interval classification, simple ratios with a factor of 5 in the numerator, such as [[5/4]], [[10/9]], and [[15/8]], are major intervals, while simple ratios with a factor of 5 in the denominator, such as [[6/5]], [[16/15]], and [[9/5]], are minor intervals. Major intervals are flat of their corresponding Pythagorean interval by 81/80; for example, 5/4 is flat of [[81/64]], the Pythagorean major third, by 81/80. Minor intervals, on the other hand are sharp of their corresponding Pythagorean interval by 81/80; for example, 6/5 is 81/80 sharp of [[32/27]], the Pythagorean minor third. A simple ratio of 5 iscan generally more [[consonant]] than the Pythagorean interval which was modified by 81/80 to reach it; for example, 5/4 is more consonant than 81/64, and 6/5 is more consonant than 32/27. [[Meantone]] tempers out 81/80 so that the Pythagorean and 5-limit thirds are equated.

			The [[octave-reduced]] 5th harmonic, the classical major third 5/4, combines nicely with the octave-reduced 1st and 3rd harmonics, being the unison and perfect fifth with ratios [[1/1]] and [[3/2]] respectively, to form the major triad [[4:5:6\|1–5/4–3/2]] (4:5:6), which consists of the 5/4 major third and the 6/5 minor third stacked on top of each other, and is [[otonal]]. Its inverse is the minor triad [[10:12:15\|1–6/5–3/2]] (1/(6:5:4) = 10:12:15), which consists of 6/5 and 5/4 stacked on top of each other, and is [[utonal]]. More extended chords can be built with these intervals; for example the [[dominant seventh chord]] with steps [[20:25:30:36\|1–5/4–3/2–9/5]] (20:25:30:36), which is widely used in classical and contemporary music.

	== See also ==		== See also ==
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	== References ==		== References ==
	* {{OEIS\|A081464}} – Numbers ''k'' such that the fractional part of (3/2)<sup>''k''</~~sup~~> ~~decreases monotonically to zero~~		<references/>
	* {{OEIS\|A267122}} – Numbers ''n'' such that (3/2)<sup>n</sup> is closer to an integer than (3/2)<sup>m</sup> for any 0 < ''m'' < ''n''

FloraC: Style

2025-05-29T10:11:01Z

Style

← Older revision		Revision as of 10:11, 29 May 2025
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	== References ==		== References ==
	* {{OEIS\|A081464}} – Numbers k such that the fractional part of (3/2)^k decreases monotonically to zero		* {{OEIS\|A081464}} – Numbers ''k'' such that the fractional part of (3/2)<sup>''k''</sup> decreases monotonically to zero
	* {{OEIS\|A267122}} – Numbers n such that ~~1.5^~~n is closer to an integer than ~~1.5^~~m for any 0 < m < n		* {{OEIS\|A267122}} – Numbers ''n'' such that (3/2)<sup>n</sup> is closer to an integer than (3/2)<sup>m</sup> for any 0 < ''m'' < ''n''

Xenllium at 23:13, 27 May 2025

2025-05-27T23:13:25Z

← Older revision		Revision as of 23:13, 27 May 2025
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	== References ==		== References ==
	* {{OEIS\|A081464}}		* {{OEIS\|A081464}} – Numbers k such that the fractional part of (3/2)^k decreases monotonically to zero
			* {{OEIS\|A267122}} – Numbers n such that 1.5^n is closer to an integer than 1.5^m for any 0 < m < n

FloraC: Expand

2025-04-02T10:49:22Z

Expand

← Older revision		Revision as of 10:49, 2 April 2025
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	\| Sound = jid_5_1_pluck_adu_dr110.mp3		\| Sound = jid_5_1_pluck_adu_dr110.mp3
	}}		}}
	'''5/1''', the '''5th harmonic''', '''pentave''' or '''quintuple''', is the [[harmonic]] past [[4/1]] and before [[6/1]]. It is two [[octave]]s above [[5/4]].		'''5/1''', the '''5th harmonic''', '''pentave''' or '''quintuple''', is the [[harmonic]] past [[4/1]] and before [[6/1]]. It is two [[octave]]s above [[5/4]], and is the basis of [[5-limit]] harmony, as many 5-limit intervals can be expressed as the difference between this and another harmonic.

	5/1 is on a list of integer harmonics that approximate closest a given stack of fifths, the error being the [[81/80\|syntonic comma]].		5/1 is on a list of integer harmonics that approximate closest a given stack of fifths, the error being the [[81/80\|syntonic comma]].

FloraC: +"quintuple"

2025-03-31T10:21:06Z

+"quintuple"

← Older revision		Revision as of 10:21, 31 March 2025
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	{{Infobox Interval		{{Infobox Interval
	\| Name = 5th harmonic, pentave		\| Name = 5th harmonic, pentave, quintuple
	\| Color name = ccy3, cocoyo 3rd		\| Color name = ccy3, cocoyo 3rd
	\| Sound = jid_5_1_pluck_adu_dr110.mp3		\| Sound = jid_5_1_pluck_adu_dr110.mp3
	}}		}}
			'''5/1''', the '''5th harmonic''', '''pentave''' or '''quintuple''', is the [[harmonic]] past [[4/1]] and before [[6/1]]. It is two [[octave]]s above [[5/4]].
	'''5/1''', the '''5th harmonic''', or ~~the~~ '''~~pentave~~''', is the [[harmonic]] past [[4/1]] and before [[6/1]]. It is two [[octave]]s above [[5/4]].

	5/1 is on a list of integer harmonics that approximate closest a given stack of fifths, the error being the [[81/80\|syntonic comma]].		5/1 is on a list of integer harmonics that approximate closest a given stack of fifths, the error being the [[81/80\|syntonic comma]].

FloraC: Expand

2024-11-30T15:34:55Z

Expand

← Older revision		Revision as of 15:34, 30 November 2024
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	}}		}}

	'''5/1''' is the 5th [[harmonic]], or the '''pentave'''; two ~~octaves~~ above [[~~5/4\|~~5/4]].		'''5/1''', the '''5th harmonic''', or the '''pentave''', is the [[harmonic]] past [[4/1]] and before [[6/1]]. It is two [[octave]]s above [[5/4]].

	5/1 is on a list of integer harmonics that approximate closest a given stack of fifths, the error being the [[81/80\|syntonic comma]].		5/1 is on a list of integer harmonics that approximate closest a given stack of fifths, the error being the [[81/80\|syntonic comma]].

	== See also ==		== See also ==
	* [[~~ED5~~]] (equal divisions of the ~~pentave)~~		* [[Ed5]] – equal divisions of the 5th harmonic

	== References ==		== References ==
	* ~~[[oeis:~~A081464]]		* {{OEIS\|A081464}}

Lériendil at 22:32, 26 August 2024

2024-08-26T22:32:48Z

← Older revision		Revision as of 22:32, 26 August 2024
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	5/1 is on a list of integer harmonics that approximate closest a given stack of fifths, the error being the [[81/80\|syntonic comma]].		5/1 is on a list of integer harmonics that approximate closest a given stack of fifths, the error being the [[81/80\|syntonic comma]].

			== See also ==
			* [[ED5]] (equal divisions of the pentave)

	== References ==		== References ==
	* [[oeis:A081464]]		* [[oeis:A081464]]

← Older revision		Revision as of 15:38, 20 March 2026
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	== See also ==		== See also ==
			* [[5/4]] – its [[octave reduced]] form
	* [[Ed5]] – equal divisions of the 5th harmonic		* [[Ed5]] – equal divisions of the 5th harmonic

	== References ==		== References ==
	<references/>		<references/>